Revenue Maximization. Sept. 25, 2018

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1 Revenue Maximization Sept. 25, 2018

2 Goal So Far: Ideal Auctions Dominant-Strategy Incentive Compatible (DSIC) b i = v i is a dominant strategy u i 0 x is welfare-maximizing x and p run in polynomial time

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4 Myerson s Lemma Still Applies In single-parameter environments, we have only one choice for building DSIC auctions: Choose a monotone allocation rule Myerson s lemma determines the payments Let s think about single-item auctions. What options are still available to us?

5 Reserve Prices Our allocation rule needs to be monotone. The only way to increase revenue is to move the jump in allocation to the right. We can move the jump by refusing to allocate the good if the maximum bid is below R. How should we choose the reserve price, R? x(b i ) b i

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sha1_base64="+ezzgvt0tlrupr5012jyx8y5hfq=">aaackxicbvdlsgmxfm3uvx1fvzdugkwpmzijgi4qfny4rmu+ofogthrbhmyyq5irytdfceovuffq1k0/ytroqlspbe7oufcm9wqxz0o7zqevw1ldw9/ib9pb2zu7e4x9g6akekmhqsmeyxzafhamokgz5tcojzaw4nakrjdtv/uaurfi3otxdn2qdatrm0q0kfxcnfbduuc7z/gaewemmeipgacmdh2fymngbwc47nm2k90rug57ihpznv8oomvnbrxm3iwuuyaax3j1ehfnqhcacqjux3vi3u2j1ixymnheoiamdeqg0dfukbbun51tosenrunhfitnerrp1n8dkqmvgoebqqyjhqpfbyr+53us3b/qpkzeiqzb5w/1e451hkex4r6tqdufg0kozoavma6jjfsbcg0tgru48jjpnpddw+8uitvkfkcehafjveiuukrvditqqieoektp6a29w0/wi/vhfc1lc1bwc4j+wpr+axbronc=</latexit> <latexit sha1_base64="+ezzgvt0tlrupr5012jyx8y5hfq=">aaackxicbvdlsgmxfm3uvx1fvzdugkwpmzijgi4qfny4rmu+ofogthrbhmyyq5irytdfceovuffq1k0/ytroqlspbe7oufcm9wqxz0o7zqevw1ldw9/ib9pb2zu7e4x9g6akekmhqsmeyxzafhamokgz5tcojzaw4nakrjdtv/uaurfi3otxdn2qdatrm0q0kfxcnfbduuc7z/gaewemmeipgacmdh2fymngbwc47nm2k90rug57ihpznv8oomvnbrxm3iwuuyaax3j1ehfnqhcacqjux3vi3u2j1ixymnheoiamdeqg0dfukbbun51tosenrunhfitnerrp1n8dkqmvgoebqqyjhqpfbyr+53us3b/qpkzeiqzb5w/1e451hkex4r6tqdufg0kozoavma6jjfsbcg0tgru48jjpnpddw+8uitvkfkcehafjveiuukrvditqqieoektp6a29w0/wi/vhfc1lc1bwc4j+wpr+axbronc=</latexit> Optimal Reserve Price: 1 Bidder With one bidder, the reserve price is a take-it-orleave-it offer: x 1 (b 1 )= ( 0 b 1 <R 1 b 1 R p 1 (b 1 )= ( R b 1 R 0 b 1 <R Give a formula for expected revenue. What information do we need?

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8 Optimal Reserve Price: >1 Bidders When does the reserve price matter? When the 2 nd highest bid is below the reserve price ER =Pr(v 2 R) v 2 + Pr(v 2 <R^ v 1 >R) R + Pr(v 1 apple R) 0 reserve is irrelevant reserve helps reserve hurts

9 Optimal Revenue Theory The following achieves optimal revenue in a singleparameter environment: Convert every agent s b i into a virtual valuation φ i, which depends on their value distribution. Run a DSIC virtual-welfare maximizing auction. Critical bid = φ -1 (critical virtual value) Virtual values can be negative. If the max virtual value is negative, don t allocate. If the critical virtual value is negative, charge φ -1 (0).

10 Why aren t we studying this? The math is hard. There s no stats prerequisite for this class. The theory is far less applicable. Needing to know every bidder s value distribution is a much stronger assumption than we ve been making so far. Optimizing reserve prices isn t very important. Bulow-Klemperer Theorem If you re interested, you can read the rest of chapters 5 & 6.

11 <latexit sha1_base64="g7awayf9lmkymgknmg07ho5yeyg=">aaad2xicjvplbtnafj3epip5niulmysivymvkrtcaokgvujflene0kpxao3hk2rue2x5xkgr8yifclhlz9jxffwc145b2iygrrj8fc+zc+6xx34scqvt+2ejady4eev2xh3z7r37dzzbww+hks5sxgcsdup0xkekh0lygry65cdjymnkh/zyp3td4sdzniory/d6kfbxrkdstasjglvevuox5aos8nlrdyrtxbaqealcke90b9w5z7lfgjuk6uzvmfbior3fc3agc67pp0lmdl1vyvp92aze4ad64lqm5dtp7prd37rcloml7hgsuscieub7+vhh5zxrvciwxip475hq+7jylev7aqfm6psk50cubbeu/dgyc9slovf2l0odo1xjjewujc4pvy4ppebmlxehhuwrzkec7q9um2usysirwrtms+v8jsyczdbovensnjzhh5jynzwspu7zp8wknndq23t2twc1coqiterv81o/3cbmwcslzifvautyir7nnnwchbww3uzxhlizouujlcwnubrn1ckswmjoajm4xutqqlqxd+q0umor4qe1yotqoly212gjte9ejhmhk0xzyzzgkywehun5zceqkwc6xgbbwspwvmazmlkm8wcw8su41yovfsp9pqfrd8/ahwf169ggj8kt0ieoeu4oyvvsiwpcmspmx+bn5hdjzhwyvhrfltrmo97zifxzxvff9ykvfa==</latexit> <latexit sha1_base64="g7awayf9lmkymgknmg07ho5yeyg=">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</latexit> <latexit sha1_base64="g7awayf9lmkymgknmg07ho5yeyg=">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</latexit> <latexit sha1_base64="g7awayf9lmkymgknmg07ho5yeyg=">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</latexit> Bulow-Klemperer Theorem E ~v F n+1 " n+1 X p i (~v) # E ~v F n " n X p i (~v) # i=1 i=1 F is some value distribution p is the ordinary second-price auction payment rule p* is the payment rule for the revenue-optimal auction on F What does this mean? In a single item auction, finding one more bidder helps more than choosing an optimal reserve price.

12 Bulow-Klemperer Proof Sketch Step 1: prove that the second price auction is revenue-optimal among single-item auctions that always allocate the good. Step 2: construct a silly n + 1 bidder auction: Run the optimal n-bidder auction among the first n agents. If the item wasn t awarded to anyone, give it to bidder n+1 for free. This auction clearly has the same expected revenue as the optimal n-bidder auction. Because it always allocates the item, step 1 proves that it can t be better than 2 nd price.